A heat engine is being designed to have a Carnot efficiency...

A heat engine is being designed to have a Carnot efficiency of 65% when operating between two heat reservoirs. (a) If the temperature of the cold reservoir is $20^{\circ} \mathrm{C},$ what must be the temperature of the hot reservoir? (b) Can the actual efficiency of the engine be equal to 65$\% ?$ Explain.

Key Concepts

Second Law of Thermodynamics

The second law of thermodynamics underpins the limits on energy conversion processes by stating that heat cannot spontaneously flow from a colder body to a hotter body. This law explains why it is impossible to achieve 100% efficiency in any heat engine and establishes that even the ideal Carnot engine, which represents a reversible process, can only achieve maximum efficiency based solely on the temperatures of the reservoirs.

Practical vs. Theoretical Efficiency

While theoretical models like the Carnot engine set an upper bound for efficiency, real-world heat engines face various irreversibilities, such as friction and non-reversible heat transfers, that reduce efficiency. Therefore, actual engine performances will always fall short of the ideal Carnot efficiency, making it a standard of performance rather than an attainable metric in practice.

Carnot Efficiency

Carnot efficiency is the theoretical maximum efficiency achievable by any heat engine operating between two fixed temperatures. It is defined by the formula ? = 1 - (T_cold/T_hot), where temperatures are measured in absolute units (Kelvin). This concept embodies the ideal performance limit, implying that no real engine can exceed this efficiency due to inherent irreversibilities.

Absolute Temperature Scale

The absolute temperature scale, typically Kelvin, is essential in thermodynamics because it begins at absolute zero, the point where molecular motion stops. Using the Kelvin scale guarantees that temperature ratios used in efficiency formulas, such as the Carnot efficiency equation, are consistent and meaningful in representing the real energy differences between reservoirs.

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